用來監控資料是否服從常態分配的偏度與峰度管制圖

The Skewness and Kurtosis Control Charts for Monitoring the Normality

童玟婷(Wen-Ting Tung)

指導教授 ： 陳慧芬

中原大學/工學院/工業工程研究所/碩士(2009年)

https://doi.org/10.6840/cycu200900308

摘要

本研究針對偏態及峰態管制圖作為主要研究研究目標，主要用來分析資料是
否有服從常態分配。雖然現今有許多X 或者R 等管制圖，但是對於服從常態分配
的能力如何實在不易得知。於是藉由偏度與峰度在常態分配下的定義給予建立管
制圖，並由此判斷資料是否為常態分配。
然而當我們設計管制圖時，必須要有樣本數及管制圖上下限的距離因素。因
此必須在一開始就先把這兩個參數先給設定好。但由於偏度與峰度為未知分配，
所以必須使用模擬方法求得管制圖上下限的距離。而在常態分配下，由於我們已
知偏度的值是為對稱分佈，故可以使用回朔近似法。但峰度為不對稱分佈，所以
我們將使用百分位數估計法來求得。
設計好偏度與峰度管制圖後，我們將探討資料是否有服從常態分配。雖然文
獻上不乏探討非常態資料的研究，但是皆未得知資料在常態分配下的偵測能力為
何，加上資料的樣本平均數之分配會受到樣本數的影響，所以難以得知樣本在常
態分配下的實際情況。因此我們將進行偏度與峰度在常態分配下的定義來評估資
料是否有服從常態分配。
因而我們先假設樣本觀察值是獨立從一組常態分配抽樣出來，接著再假設樣
本觀察值也是獨立從另一組Johnson 分配抽樣出來。由於Johnson 分配家族可涵
蓋所有可能的偏度與峰度值，在管制圖的應用上範圍很廣。因而將此兩種分配所
產生出來的資料作為比較，但是缺點是型一或型二誤差必須藉由平均連串長度的
模擬實驗求得。我們將由實驗得知，偏度與峰度的管制圖在常態與非常態分配的
偵測能力。

關鍵字

偏度管制圖 ； 峰度管制圖 ； 平均連串長度 ； 回朔近似法 ； 百分位數估計

並列摘要

This thesis major research is skewness and kurtosis control charts. We want to
analyses the data if following normal distribution. Now the data can use many control
charts to estimate like ¯X or R control charts. But we do not know the data if following
normal distribution how is the ability. So we use skewness control chart and kurtosis
control chart to monitor the data. Because the mean will changes in ¯X control chart
that we assess the data if following normal distribution.
When we design the control charts, we have two parameters must given before
establish skewness control chart and kurtosis control chart. The parameters are sample
size and the distances from the centerline to the control limits. But we do not
know the distribution of skewness and kurtosis, we must simulate the parameter for
the distances from the centerline to the control limits. Then use the definition of
skewness and kurtosis in normal distribution. And we know the skewness is symmrtric
that we use retrospective approximation (RA) compute the distances from the
centerline to the control limits. The kurtosis is asymmetric that we use overlapping
batch quantile(OBQ) compute the distances from the centerline to the control limits.
We will use the skewness control chart and kurtosis control chart to monitor the
data if following normal distribution. Although some nonnormal literature exists,
the assumption is made on the distribution of sample average, which depends on the
unknown sample size. So we must use skewness and kurtosis control charts to monitor
the data if following normal distribution.
We assume that the data are sampled independently from the normal distribution,
and the other are sampled independently from Johnson distribution. The Johnson
distribution is general in that it can be modeled to fit all possible values of the
skewness and kurtosis. We will compare the data in skewness and kurtosis control
charts. But Type I and Type II error probabilities is difficult to compute and need
to be estimated via simulation experiment. We estimate the Type I and Type II
error probabilities by average run length (ARL). We know the ability in skewness
and kurtosis control charts with the experiment.

並列關鍵字

Kurtosis Control Chart ； Skewness Control Chart ； Average Run Length ； Retrospective Approximation ； Overlapping Batch Quantile

參考文獻

[2] Fisher, R. A. (1930). The moments of the distribution for normal samples of measures

of departures from normality, Proceedings of the Royal Society of London

A130, 16-28.

moments of the distribution of b2 in sample form a normal population, Biometrika

[5] Stuart,A. and Ord, J. K. (1987). Kendall’s Advanced Theory of Statistics,